BZMVdM algebras and stonian MV-algebras (applications to fuzzy sets and rough approximations)

✍ Scribed by G. Cattaneo; R. Giuntini; R. Pilla

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 205 KB
• Volume: 108
• Category: Article
• ISSN: 0165-0114
• DOI: 10.1016/s0165-0114(97)00328-x

✦ Synopsis

The natural algebraic structure of fuzzy sets suggests the introduction of an abstract algebraic structure called de Morgan BZMV-algebra (BZMV dM -algebra). We study this structure and sketch its main properties. A BZMV dM -algebra is a system endowed with a commutative and associative binary operator ⊕ and two unusual orthocomplementations: a Kleene orthocomplementation (" ) and a Brouwerian one (∼). As expected, every BZMV dM -algebra is both an MV-algebra and a distributive de Morgan BZ-lattice. The set of all ∼-closed elements (which coincides with the set of all ⊕ -idempotent elements) turns out to be a Boolean algebra (the Boolean algebra of sharp or crisp elements). By means of " and ∼, two modal-like unary operators ( for necessity and for possibility) can be introduced in such a way that (a) (resp., (a)) can be regarded as the sharp approximation from the bottom (resp., top) of a. This gives rise to the rough approximation ( (a); (a)) of a. Finally, we prove that BZMV dM -algebras (which are equationally characterized) are the same as the Stonian MV-algebras and a ÿrst representation theorem is proved.